Determination of a time-dependent coefficient in a non-linear hyperbolic equation with non-classical boundary condition

The non-linear hyperbolic equation is used to model many non-linear phenomena. In this paper, we consider an initial boundary value problem for non-linear hyperbolic equation. We determine a time-dependent coefficient multiplying non-linear term by using an additional condition, and prove the existence and uniqueness theorem for small times. We also propose a numerical scheme to solve the inverse problem for non-linear hyperbolic equation, and give test examples for sine, quadratic and cubic non-linearity.